1. Chap 3 Interpolating Values
Animation(U), Chap 3, Interpolating Values

2. Outline Interpolating/approximating curves
Controlling the motion of a point along a curve
Interpolation of orientation
Working with paths
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

3. Interpolation between key frames
Parameters to be interpolated
Position of an object
Normal
Joint angle between two joints
Transparency attribute of an object
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

4. Interpolation between key frames
Interpolation between frames is not trivial
Appropriate interpolating function
Parameterization of the function
Maintain the desired control of the interpolated values over time
Example
(-5,0,0) at frame 22, (5,0,0) art frame 67
Stop at frame 22 and accelerate to reach a max speed by frame 34
Start to decelerate at frame 50 and come to a stop at frame 67
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

5. Interpolation between key frames
desired result
undesired result
Need a smooth interpolation with user control
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

6. Interpolation between key frames
Solution
Generate a space curve
Distribute points evenly along curve
Speed control: vary points temporally B
Time = 10 A
Time = 0 C
Time = 35 D
Time = 60
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

7. Interpolation functions
Interpolation vs. approximation
Interpolation
Hermite, Catmull-Rom
approximation
Bezier, B-spline
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

8. Interpolation functions
Complexity => computational efficiency
Polynomials
Lower than cubic
No inflection point, may not fit smoothly to some data points
Higher than cubic
Doesn’t provide any significant advantages, costly to evaluate
Piecewise cubic
Provides sufficient smoothness
Allows enough flexibility to satisfy constraints such as end-point position and tangents
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

9. Interpolation functions
Continuity within a curve
Zero-order
First order (tangential)
Suffices for most animation applications
Second order
Important when dealing with time-distance curve
Continuity between curve segments
Hermite, Catmull-Rom, cubic Bezier provide first order continuity between segments
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

10. Interpolation functions
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

11. Continuity position (0th order) tangent (1st order) curvature
(2nd order)
none
At junction of two circular arcs
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

12. Interpolation functions
Global vs. local control
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

13. Types of Curve Representation
Explicit y = f(x)
Good for generate points
For each input, there is a unique output
Implicit f(x,y) = 0
Good for testing if a point is on a curve
Bad for generating a sequence of points
Parametric x = f(u), y = g(u)
Good for generating a sequence of points
Can be used for multi-valued function of x
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

14. Example: Representing Unit Circle
Cannot be represented explicitly as a function of x
Implicit form: f(x,y)=x2+y2=1
Parametric form: x=cos(u), y=sin(u), 0<u<2π
f>0
f=0
f<0 u
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

15. More on 3-D Parametric Curves
Parametric form: P(u) = (Px(u), Py(u), Pz(u))
x = Px(u), y = Py(u), z = Pz(u)
u=1/3
u=2/3
u=0.0
u=1.0
Space-curve
P = P(u) <=u<=1.0
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

16. Polynomial Interpolation
An n-th degree polynomial fits a curve to n+1 points
Example: fit a second degree curve to three points
y(x)= a x2 + b x + c
points to be interpolated (x1, y1), (x2, y2), (x3, y3)
solve for coefficients (a, b, c): 3 linear equations, 3 unknowns
called Lagrange Interpolation
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

17. Polynomial Interpolation (cont.)
Result is a curve that is too wiggly, change to any control point affects entire curve (nonlocal) – this method is poor
We usually want the curve to be as smooth as possible
minimize the wiggles
high-degree polynomials are bad
Higher degree, higher the wiggles
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

18. Composite Segments Divide a curve into multiple segments
Represent each in a parametric form
Maintain continuity between segments
position
Tangent (C1-continuity vs. G1-continuity)
curvature
P1(u)
P2(u)
P3(u)
Pn(u)
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

19. Splines: Piecewise Polynomials
A spline is a piecewise polynomial - many low degree polynomials are used to interpolate (pass through) the control points
Cubic polynomials are the most common
lowest order polynomials that interpolate two points and allow the gradient at each point to be defined - C1 continuity is possible
Higher or lower degrees are possible, of course
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

20. A Linear Piecewise Polynomial
The simple form for interpolating two points. p1 p2 u
Each segment is of the form: (this is a vector equation) 1 u 1
Two basis (blending) functions
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

21. Hermite Interpolation
Hermite Curves—cubic polynomial for two points
Hermite interpolation requires
Endpoint positions
derivatives at endpoints
To create a composite curve, use the end of one as the beginning of the other and share the tangent vector
control points/knots
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

22. Hermite Curve Formation
Cubic polynomial and its derivative
Given Px(0), Px(1), P’x(0), P’x(1), solve for a, b, c, d
4 equations are given for 4 unknowns
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

23. Hermite Curve Formation (cont.)
Problem: solve for a, b, c, d
Solution:
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

24. Hermite Curves in Matrix Form
ith segment in
composite curves
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

25. Blending Functions of Hermite Splines
Each cubic Hermite spline is a linear combination of 4 blending functions
geometric information
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

26. Composite Hermite Curve
Continuity between connected segments is ensured by using ending tangent vector of one segment as the beginning tangent vector of the next.
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

27. Bezier Curves Similar to Hermite Curve
Instead of endpoints and tangents, four control points are given
points P1 and P4 are on the curve
points P2 and P3 are used to control the shape
p1 = P1, p2 = P4,
p1' = 3(P2-P1), p2' = 3(P4 - P3)
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

28. Bezier Curves Another representation
Blend the control point position using Bernstein polynomials
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

29. Bezier Curves Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

30. Composite Bezier Curves
How to control the continuity between adjacent Bezier segment?
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

31. De Casteljau Construction of Bezier Curves
How to derive a point on a Bezier curve?
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

32. Bezier Curves (cont.) Variant of the Hermite spline
basis matrix derived from the Hermite basis (or from scratch)
Gives more uniform control knobs (series of points) than Hermite
Scale factor (3) is chosen to make “velocity” approximately constant
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

33. Catmull-Rom Splines
With Hermite splines, the designer must arrange for consecutive tangents to be collinear, to get C1 continuity. Similar for Bezier. This gets tedious.
Catmull-Rom: an interpolating cubic spline with built-in C1 continuity.
Compared to Hermite/Bezier: fewer control points required, but less freedom.
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

34. Catmull-Rom Splines (cont.)
Given n control points in 3-D: p1, p2, …, pn,
Tangent at pi given by s(pi+1 – pi-1) for i=2..n-1, for some s.
Curve between pi and pi+1 is determined by pi-1, pi, pi+1, pi+2.
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

35. Catmull-Rom Splines (cont.)
Given n control points in 3-D: p1, p2, …, pn,
Tangent at pi given by s(pi+1 – pi-1) for i=2..n-1, for some s
Curve between pi and pi+1 is determined by pi-1, pi, pi+1, pi+2
What about endpoint tangents? (several good answers: extrapolate, or use extra control points p0, pn+1)
Now we have positions and tangents at each knot – a Hermite specification.
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

36. Catmull-Rom Spline Matrix
Derived similarly to Hermite and Bezier
s is the tension parameter; typically s=1/2
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

37. Catmull-Rom Spline What about endpoint tangents? Provided by users
Several good answers: extrapolate, or use extra control points p0, pn+1
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

38. Catmull-Rom Spline Example Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

39. Catmull-Rom Spline Drawback
An internal tangent is not dependent on the position of the internal point relative to its two neighbors
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

40. p(0) = P2, p’(0) = s(P3-P1), p(1) = P3, p’(1) = s(P4-P2)
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

41. Splines and Other Interpolation Forms
See Computer Graphics textbooks
Review
Appendix B.4 in Parent
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

42. Now What? We have key frames or points
We have a way to specify the space curve
Now we need to specify velocity to traverse the curve
Speed Curves
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

43. Speed Control
The speed of tracing a curve needs to be under the direct control of the animator
Varying u at a constant rate will not necessarily generate P(u) at a constant speed.
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

44. Non-uniformity in Parametrization
Generally, equally spaced samples in parameter space are not equally spaced along the curve B
Time = 10 A
Time = 0 C
Time = 35 D
Time = 60
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

45. Arc Length Reparameterization
To ensure a constant speed for the interpolated value, the curve has to be parameterized by arc length (for most applications)
Computing arc length
Analytic method (many curves do not have, e.g., B-splines)
Numeric methods
Table and differencing
Gaussian quadrature
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

46. Arc Length Reparameterization
Space curve vs. time-distance function
Relates time to the distance traveled along the curve, i.e.,
Relates time to the arc length along the curve
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

47. Arc Length Reparameterization
Given a space curve with arc length parameterization
Allow movement along the curve at a constant speed by stepping at equal arc length intervals
Allows acceleration and deceleration along the curve by controlling the distance traveled in a given time interval
Problems
Given a parametric curve and two parameter values u1 and u2, find arclength(u1,u2)
Given an arc length s, and parameter value u1, find u2 such that arclength(u1,u2) = s
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

48. Arc Length Reparameterization
Converting a space curve P(u) to a curve with arc-length parameterization
Find s=S(u), and u=S-1(s)=U(s)
P*(s)=P(U(s))
Analytic arc-length parameterization is difficult or impossible for most curves, e.g., B-spline curve cannot be parameterized with arc length.
Approximate arc-length parameterization
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

49. Forward Differencing
Sample the curve at small intervals of the parameter
Compute the distance between samples
Build a table of arc length for the curve u
Arc Length
0.0
0.00
0.1
0.08
0.2
0.19
0.3
0.32
0.4
0.45 …
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

50. Arc Length Reparameterization Using Forward Differencing
Given a parameter u, find the arc length
Find the entry in the table closest to this u
Or take the u before and after it and interpolate arc length linearly u
Arc Length
0.0
0.00
0.1
0.08
0.2
0.19
0.3
0.32
0.4
0.45 …
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

51. Arc Length Reparameterization Using Forward Differencing
Given an arc length s and a parameter u1, find the parametric value u2 such that arclength(u1, u2)=s
Find the entry in the table
closest to this u using
binary search
Or take the u before and after
it and interpolate linearly u
Arc Length
0.0
0.00
0.1
0.08
0.2
0.19
0.3
0.32
0.4
0.45 …
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

52. Arc Length Reparameterization Using Forward Differencing
Easy to implement, intuitive, and fast
Introduce errors
Super-sampling in forming table
1000 equally spaced parameter values ebtries in each interval
Better interpolation
Adaptive forward differencing
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

53. Arc Length Reparameterization Using Adaptive forward differencing
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

54. Arc Length Reparameterization Using Numerical Integral
Arc length integral
Simpson’s and trapezoidal integration using evenly spaced intervals
Gaussian quadrature using unevenly spaced intervals
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

55. Speed Control
Given a arc-length parameterized space curve, how to control the speed at which the curve is traced?
By a speed-control functions that relate an equally spaced time interval to arc length
Input time t, output arc length: s=S(t)
Linear function: constant speed control
Most common: ease-in/ease-out
Smooth motion from stopped position, accelerate, reach a max velocity, and then decelerate to a stop position
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

56. Speed Control Function
Relates an equally spaced time interval to arc length
Input time t, output arc length: s=S(t)
Normalized arc length parameter
s=ease(t)
Start at 0,
slowly increase
in value and gain
speed until the
middle value and
then decelerate as
it approaches to 1.
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

57. Constant Velocity Speed Curve
Moving at 1 m/s if meters and seconds are the units
Too simple to be what we want
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

58. Distance Time Function and Speed Control
We have a space curve p=P(u) and a speed control function s=S(t).
For a given t, s=S(t)
Find the corresponding value u=U(s) by looking up an arc length table for a given s
A point on the space curve with parameter u
p=P(U(S(t)))
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

59. Speed Control u* s* t* … u Arc Length 0.0 0.00 0.1 0.08 0.2 0.19 0.3
Arc Length Table u
Arc Length
0.0
0.00
0.1
0.08
0.2
0.19
0.3
0.32
0.4
0.45 …
Speed Control Curve u* s* t*
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

60. Distance Time Function
Assumptions on distance time function
The entire arc length of the curve is to be traversed during the given total time
Additional optional assumptions
The function should be monotonic in t
The function should be continuous
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

61. Ease-in/Ease-out
Most useful and most common ways to control motion along a curve
(distance)
Time
Arc length
Equally spaced samples in time specify arc length required for that frame
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

62. Ease-in/Ease-out Sine Interpolation-1 1
-p/2
p/2
Arc Length
Time
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

63. Ease-in/Ease-out Sine Interpolation
The speed of s w.r.t. t is never constant over an interval but rather is always accelerating or decelerating.
Zero derivatives at t=0 and t=1 indicating smooth accelerating and deceleration at end points.
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

64. Ease-in/Ease-out Piecing Curves Together for Ease In/Out
Arc Length
Linear segment
Time
Sinusoidal segments
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

65. Ease-in/Ease-out Piecing Curves Together for Ease In/Out
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

66. Ease-in/Ease-out Integrating to avoid the sine function
acceleration
time t1 t2 A D
velocity
time t1 t2 v
Find v such that area = 1
arc length
V1 = V2  A*t1 = - D*(1-t2)
time
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

67. Ease-in/Ease-out Integrating to avoid the sine function
To avoid the transcendental function evaluation while providing a constant speed interval between ease-in and ease-out intervals
Acceleration-time curve
Velocity-time curve
Velocity-time curve is defined by integral of the acceleration-time curve
Distance-time curve is defined by integral of the velocity-time curve
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

68. Ease-in/Ease-out Integrating to avoid the sine function
In ease-in/ease-out function
Velocity starts out at 0 and ends at 0
V1 = V2 inplies A*t1 = - D*(1-t2)
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

69. Ease-in/Ease-out Integrating to avoid the sine function
Total distance = 1 implies
1 = ½ V0 t + V0 (t2-t1)+1/2 V0 (1-t2)
Users specifies any two of three variables t1, t2, and V0, and the system can solve for the third
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

70. Ease-in/Ease-out Integrating to avoid the sine function
The result is Parabolic ease-in/ease-out
More flexible than the sinusoidal ease-in/ease-out
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

71. Videos Bunny (Blue Sky Studios, 1998)
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

72. Review - Quaternions Similar to axis-angle representations
4-tuple of real numbers
q=(s, x, y, z) or [s, v]s is a scalar; v is a vector
The quaternion for rotating an angle about an axis (an axis-angle rotation):
a=(ax, ay, az) Y q X
If a is unit length, then q will be also Z
Animation(U), Chap 2, Tech. Background
CS, NCTU, J. H. Chuang

73. Review - Quaternions If a is unit length, then q will be also
Animation(U), Chap 2, Tech. Background
CS, NCTU, J. H. Chuang

74. Review - Quaternions
Rotating some angle around an axis is the same as rotating the negative angle around the negated axis
Animation(U), Chap 2, Tech. Background
CS, NCTU, J. H. Chuang

75. Review - Quaternion Math
Addition
Multiplication
Multiplication is associative but not commutative
Animation(U), Chap 2, Tech. Background
CS, NCTU, J. H. Chuang

76. Review - Quaternion Math (cont.)
Multiplicative identity: [1, 0,0,0]
Inverse
Normalization for unit quaternion
Animation(U), Chap 2, Tech. Background
CS, NCTU, J. H. Chuang

77. Review – Rotating Vectors Using Quaternion
A point in space, v, is represented as [0, v]
To rotate a vector v using quaternion q
Represent the vector as v = [0, v]
Represent the rotation as a quaternion q
Using quaternion multiplication
The proof isn’t that hard
Note that the result v’ always has zero scalar value
Animation(U), Chap 2, Tech. Background
CS, NCTU, J. H. Chuang

78. Review - Compose Rotations
Rotating a vector v by first quaternion p followed by a quaternion q is like rotation using qp
Animation(U), Chap 2, Tech. Background
CS, NCTU, J. H. Chuang

79. Review - Compose Rotations
To rotate a vector v by quaternion q followed by its inverse quaternion q-1
Animation(U), Chap 2, Tech. Background
CS, NCTU, J. H. Chuang

80. Review - Quaternion Interpolation
A quaternion is a point on a 4D unit sphere
Unit quaternion: q=(s,x,y,z), ||q|| = 1
Form a subspace: a 4D sphere
Interpolating quaternion means moving between two points on the 4D unit sphere
A unit quaternion at each step – another point
on the 4D unit sphere
Move with constant angular velocity
along the greatest circle between the
two points on the 4D unit sphere
Animation(U), Chap 2, Tech. Background
CS, NCTU, J. H. Chuang

81. Review - Linear Interpolation
Linear interpolation generates unequal spacing of points after projecting to circle
Animation(U), Chap 2, Tech. Background
CS, NCTU, J. H. Chuang

82. Review – Spherical Linear Interpolation (slerp)
Want equal increment along arc connecting two quaternion on the spherical surface
Spherical linear interpolation (slerp)
Normalize to regain unit quaternion
Animation(U), Chap 2, Tech. Background
CS, NCTU, J. H. Chuang

83. Review – Spherical Linear Interpolation (slerp)
Animation(U), Chap 2, Tech. Background
CS, NCTU, J. H. Chuang

84. Review – Spherical Linear Interpolation (slerp)
Animation(U), Chap 2, Tech. Background
CS, NCTU, J. H. Chuang

85. Review – Spherical Linear Interpolation (slerp)
Recall that q and –q represent same rotation
What is the difference between:
Slerp(u, q1, q2) and Slerp(u, q1, -q2) ?
One of these will travel less than 90 degrees while the other will travel more than 90 degrees across the sphere
This corresponds to rotating the ‘short way’ or the ‘long way’
Usually, we want to take the short way, so we negate one of them if their dot product is < 0
Animation(U), Chap 2, Tech. Background
CS, NCTU, J. H. Chuang

86. Review – Spherical Linear Interpolation (slerp)
If we have an intermediate position q2, the interpolation from q1-->q2-->q3 will not necessarily follow the same path as the interpolation from q1 to q3.
Animation(U), Chap 2, Tech. Background
CS, NCTU, J. H. Chuang

87. Interpolating a Series of Quaternions
As linear interpolation in Euclidean space, we can have first order discontinuity problem when interpolating a series of orientations.
Need a cubic curve interpolation to maintain first order continuity in Euclidean space
Similarly, slerp can have 1st order discontinuity
We also need a cubic curve interpolation in 4D spherical space for 1st order continuity
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

88. Interpolating a Series of Quaternions
Interpolated points […, pn-1, pn, pn+1, ….] in 3D
Between each pair of points, two control points will be constructed. e.g., For pn
an : the one intermediate after pn
bn : the one immediately before pn
pn+(pn-pn-1)
Take average
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

89. Interpolating a Series of Quaternions
Interpolated points […, pn-1, pn, pn+1, ….] in 3D
Between each pair of points, two control points will be constructed. e.g., For pn
an : the one intermediate after pn
bn : the one immediately before pn
bn =pn+(pn-an)
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

90. Interpolating a Series of Quaternions
Interpolated points […, pn-1, pn, pn+1, ….] in 3D
End conditions
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

91. Interpolating a Series of Quaternions
Interpolated points […, pn-1, pn, pn+1, ….] in 3D
Between any 2 interpolated points pn and pn+1, a cubic Bezier curve segment is defined by pn, an, bn+1, and pn+1
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

92. Interpolating a Series of Quaternions
How to evaluate a point on Bezier curve?
De Casteljau Construction
Constructing Bezier curve by multiple linear interpolation
u=1/3
p(1/3)
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

93. Interpolating a Series of Quaternions
Animation: See
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

94. Interpolating a Series of Quaternions
Bezier interpolation on 4D sphere?
How are control points generated?
How are cubic Beziers defined?
Control points are automatically generated as it is not intuitive to manually adjust them on a 4D sphere
Derive Bezier curve points by iteratively subdivision s on pn, an, bn+1, and pn+1
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

95. Interpolating a Series of Quaternions Connecting Segments on 4D Sphere
Automatically generating interior (spherical) control points
qn’
2. Bisect the span an bn qn
qn+1
qn-1
1. Double the arc
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

96. Interpolating a Series of Quaternions De Casteljau Construction on 4D Sphere
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

97. Path Following Issues about path following
Arc length parameterization and speed control
Changing the orientation
Smoothing a path
Path along a surface
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

98. Orientation Along a Curve
A local coordinate system (u, v, w) is defined for an object to be animated
Position along the path P(s), denoted as POS
w-axis: the direction the object is facing
v-axis: up vector
u-axis: perpendicular to w and v
(u, v, w): a right-handed coordinate system
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

99. Orientation Along a Curve
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

100. Orientation Along a Curve Frenet Frame
Frenet frame: a moving coordinate system determined by curve tangent and curvature
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

101. Orientation Along a Curve Frenet Frame
Problems
No concept of “up” vector
Is undefined in segments that has no curvature; i.e., P”(u)=0
Can be dealt with by interpolating a Frenet frame from the Frenet frames at segment’s boundary, which differ by only a rotation around w
A discontinuity in the curvature vector
Frenet frame has a discontinuous jump in orientation
Resulting motions are usually too extreme and not natural looking
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

102. Orientation Along a Curve Frenet Frame
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

103. Orientation Along a Curve Frenet Frame
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

104. Orientation Along a Curve Camera Path Following
COI: center of interest
A fixed point in the scene
Center point of an object in the scene
Points along the path itself
Points on a separate path in the scene
Points on the interpolation between points in the scene
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

105. Smoothing a Path with Linear Interpolation
Two points on either sides of a point are averaged, and this point is averaged with the original point.
Repeated application
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

106. Smoothing a Path with Linear Interpolation
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

107. Smoothing a Path with Cubic Interpolation
Four points on either sides of a point are fitted by a cubic curve, and the midpoint of the curve is averaged with the original point.
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

108. Smoothing a Path with Cubic Interpolation
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

109. Smoothing a Path with Cubic Interpolation
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

110. Determine a Path on a Surface
If one object is to move across the surface of another object, we need to specify a path across the surface.
Given start and destination points
Find a shortest path between the points – expensive
Determine a plane that contain two points and is generally perpendicular to the surface (averaged normal of two points)
Intersection of the plane with the surface (mesh)
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

111. Determine a Path on a Surface
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang

112. Determine a Path on a Parametric Surface
Given start and destination points on parametric domain
Define the straight line connecting two points in the parametric space
The curve on the surface corresponding to this line
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang